Finitely presentable morphisms in exact sequences

Author's Department

Mathematics & Actuarial Science Department

Document Type

Research Article

Publication Title

Theory and Applications of Categories

Publication Date



Let K be a locally finitely presentable category. If K is abelian and the sequence, we show that 1) K is finitely generated ⇔ c is finitely presentable; 2) k is finitely presentable ⇔ C is finitely presentable. The "⇐" directions fail for semi-abelian varieties. We show that all but (possibly) 2)(⇐) follow from analogous properties which hold in all locally finitely presentable categories. As for 2)(⇐), it holds as soon as K is also co-homological, and all its strong epimorphisms are regular. Finally, locally finitely coherent (resp. noetherian) abelian categories are characterized as those for which all finitely presentable morphisms have finitely generated (resp. presentable) kernel objects. © Michel Hébert, 2010.

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